In this work, we consider a distributed variant of (REF ) where each subsystem can only exchange information with a small subset of subsystems. Specifically, this communication constraint is encoded as a graph \(\mathcal {G}= \lbrace \mathcal {V}, \mathcal {E}\rbrace \) , where \(\mathcal {V}= \lbrace v_{1},\ldots ,v_{N}\rbrace \)  is the set of \(N\)  components (nodes) and where \(\mathcal {E}\subseteq \mathcal {V}\times \mathcal {V}\)  is the set of the corresponding interconnections (edges). It is assumed that the graph is undirected, i.e. \((v_{i},v_{j}) \in \mathcal {E}\)  if and only if \((v_{j},v_{i}) \in \mathcal {E}\) . As described in the introduction, general information exchange constraints can lead to non-convex optimal control problems [1]}. However, as we show later, under suitable graph symmetry assumptions on the dynamics matrices \(A, B\)  and the cost matrices \(Q, R\) , the optimal centralized controller admits a distributed message passing implementation allowing for a principled tradeoff between communication complexity and controller performance.
